Stable rank of corner rings

Details Stable rank of corner rings Stable rank of corner rings

2003-09-06

arxiv.org/abs/math/0309116v1

Mathematics

Rings and Algebras

Scientific paper

B. Blackadar recently proved that any full corner $pAp$ in a unital C*-algebra $A$ has K-theoretic stable rank greater than or equal to the stable rank of $A$. (Here $p$ is a projection in $A$, and fullness means that $ApA=A$.) This result is extended to arbitrary (unital) rings $A$ in the present paper: If $p$ is a full idempotent in $A$, then $sr(pAp) \geq sr(A)$. The proofs rely partly on algebraic analogs of Blackadar's methods, and partly on a new technique for reducing problems of higher stable rank to a concept of stable rank one for skew (rectangular) corners $pAq$. The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if $B$ is isomorphic to the endomorphism ring of a finitely generated projective generator $P_A$ which can be generated by $n$ elements, then $sr(A) \leq n{\cdot}sr(B)-n+1$.

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